In recent years, a signal analysis filter bank which divides a signal into a plurality of band signals and a signal synthesis filter bank which reproduces an original signal by synthesizing the band signals have drawn attention as signal analyzing and signal synthesizing means which realizes high efficiency coding due to sub-band coding of audio signals and image signals. In particular, in such a bandwidth extension technique of an audio signal for realizing wideband reproduction with a few amount of information, a complex exponential modulation filter bank has been drawn attention. The complex exponential modulation filter bank has such an advantage that there occurs no aliasing even if gain is changed with respect to each band, unlike a cosine modulation filter bank which has been used in MPEG (Moving Picture Experts Group)-1 audio etc. Therefore, it is usable as various digital equalizers. However, the complex exponential modulation filter bank handles band signals with complex numbers, and therefore, it has such a drawback that quantity of arithmetic operation is large, as compared to the cosine modulation filter bank which handles the band signals with real numbers.
For example, in AAC+SBR (Spectral Band Replication) which is a bandwidth extension technique of MPEG (Moving Picture Experts Group) AAC (Advanced Audio Coding), the complex exponential modulation filter bank is used. This complex exponential modulation filter bank is disclosed in ISO/IEC 14496-3:2001, Information technology—Coding of audio-visual objects—Part 3: Audio, ISO/IEC JTC1/SC29/WG11/N5570, March 2003 (Text of ISO/IEC 14496-3:2001/FDAM1, Bandwidth Extension).
Hereinafter, a conventional signal analyzing and signal synthesizing method of a complex exponential modulation filter bank will be described. FIG. 12 is a block diagram which shows configurations of a signal analysis filter bank and a signal synthesis filter bank.
In FIG. 12, signal analysis filter bank 1201 includes sampling frequency K times interpolator 1203 and M analysis band-pass filters 1204 and M decimators 1205. The signal synthesis filter bank 1202 includes M interpolators 1206, M synthesis band-pass filters 1207, adder 1208, and sampling frequency 1/L times decimator 1209. The analysis band-pass filter 1204 and the synthesis band-pass filter 1207 are paired each other. Here, K and L are a divisor of number of bands M, and a positive integers including 1. Meanwhile, it is also possible, in FIG. 12, to realize a configuration which does not have sampling frequency K times interpolator 1203, or sampling frequency 1/L times decimator 1209 (i.e., such a configuration that a value of K or L is 1).
Firstly, an operation of the signal analysis filter bank 1201 will be described. In the sampling frequency K times interpolator 1203, by inserting (K−1) zero data with respect to each data, to an input signal with sampling frequency fs, sampling frequency is elevated by K times, to become Kfs. Next, this signal becomes band-pass signals by analysis band-pass filter 1204 which divides an entire band into M bands with equal bandwidth, and (M−1) data are removed with respect to each M data, by decimator 1205, and 1 piece data is outputted, and thereby, it is converted into a band signal with sampling frequency fsK/M, and outputted. Since sampling frequency of an input signal is set to K times, a band signal in (M/K)-th band or above is zero.
Next, an operation of signal synthesis filter bank 1202 will be described. A band signal with sampling frequency fsK/M which was outputted from signal analysis filter bank 1201 is used as an input, and (M−1) zero data is inserted by interpolator 1206 with respect to each 1 piece of data, and thereby, sampling frequency is elevated to Kfs which is M times. This signal is converted into band-pass signals by M synthesis band-pass filters 1207 with equal bandwidth, and thereafter, they are synthesized by adder 1208, so that a signal with sampling frequency Kfs is reproduced. Next, by sampling frequency 1/L times decimator 1209, (L−1) pieces of data are removed with respect to each L pieces of data, so that a signal with sampling frequency fsK/L is outputted.
In sub-band coding of audio signals and image signals, information compression is carried out between the signal analysis filter bank and the signal synthesis filter bank, through the use of deviation of distribution of a frequency direction of a band signal, and an auditory characteristic or a visual characteristic of humans to realize a high efficiency coding.
As described in international publication number WO 02/080362 A1 document, the complex exponential modulation filter bank is configured by modulating a prototype filter with complex exponentials. By this patent document, assuming that a filter coefficient of a linear phase non-recursive type prototype filter is h(n) (0≦n≦N, N is a filter order), a filter coefficient ha(k, n) of n sample in k-th band of a complex exponential modulation signal analysis filter bank is given by (formula 1) (j is imaginary unit, and A is phase for signal analysis).ha(k,n)=Kh(Kn)exp(jπ(k+0.5)(2Kn+A)/(2M))  (formula 1)
Therefore, when values of filter coefficients of first and last prototype filters are set to be zero, assuming that an input signal at sampling time n of a signal analysis filter bank is x(n), complex band output signal X(k, mM/K) at sample time mM/K in k-th band (0≦k≦M/K−1) is give by (formula 2).
                              X          ⁡                      (                          k              ,                              mM                /                K                                      )                          =                  K          ⁢                                    ∑                              n                =                0                                                              N                  /                  K                                -                1                                      ⁢                                                  ⁢                                          h                ⁡                                  (                  Kn                  )                                            ⁢                                                          ⁢                              exp                (                                                                  ⁢                                                      jπ                    ⁡                                          (                                              k                        +                        0.5                                            )                                                        ⁢                                                            (                                                                        2                          ⁢                          Kn                                                +                        A                                            )                                        /                                                                               ⁢                                                                 (                                      2                    ⁢                    M                                    )                                )                            ⁢                                                          ⁢                              x                ⁡                                  (                                                            mM                      /                      K                                        -                    n                                    )                                                                                        (                  formula          ⁢                                          ⁢          2                )            
If (formula 2) is calculated directly, quantity of arithmetic operation becomes large, and therefore, as shown below, a complex exponential modulation signal analysis filter bank with reduced quantity of arithmetic operation is used. FIG. 13 is a flow chart which shows processing steps of a conventional analyzing method of a complex exponential modulation signal analysis filter bank. In step 1301, intermediate signal w(n) is calculated by (formula 3) from input signal x(n) at sampling time n.
                                          w            ⁡                          (              n              )                                =                                    ∑                              i                =                0                                                              N                  /                                      (                                          2                      ⁢                      M                                        )                                                  -                1                                      ⁢                                                  ⁢                                                            (                                      -                    1                                    )                                i                            ⁢                              x                ⁡                                  (                                                            mM                      /                      K                                        -                                          2                      ⁢                                              Mi                        /                        K                                                              -                    n                                    )                                            ⁢                              h                ⁡                                  (                                                            2                      ⁢                      Mi                                        +                    Kn                                    )                                                                    ⁢                                  ⁢                                  ⁢                  (                      0            ≦            n            ≦                                          2                ⁢                                  M                  /                  K                                            -              1                                )                                    (                  formula          ⁢                                          ⁢          3                )            
Next, in step 1302, complex band output signal X (k, mM/K) at sampling time mM/K in k-th band (0≦k≦M/K−1) is calculated by (formula 4) (A is phase for signal analysis) from intermediate signal w(n).
                                          X            ⁡                          (                              k                ,                                  mM                  /                  K                                            )                                =                      K            ⁢                                          ∑                                  n                  =                  0                                                                      2                    ⁢                                          M                      /                      K                                                        -                  1                                            ⁢                                                          ⁢                                                w                  ⁡                                      (                    n                    )                                                  ⁢                                  exp                  ⁡                                      (                                          j                      ⁢                                                                                          ⁢                                              π                        ⁡                                                  (                                                      k                            +                            0.5                                                    )                                                                    ⁢                                                                        (                                                                                    2                              ⁢                              Kn                                                        +                            A                                                    )                                                ⁢                                                                                                  /                                                  (                                                      2                            ⁢                            M                                                    )                                                                                      )                                                                                      ⁢                                  ⁢                                  ⁢                  (                      0            ≦            k            ≦                                          M                /                K                            -              1                                )                                    (                  formula          ⁢                                          ⁢          4                )            
For example, pages 60 and 62 in the above-described ISO/IEC document describe an example of a complex exponential modulation signal analysis filter bank for such a case that number of bands M is 64, and filter order N of a prototype filter is 640, and scaling factor k of up-sampling is 2, and phase A of signals is −1. In this regard, however, this document uses c(n) which was calculated by (formula 5) from h(n), in lieu of filter coefficient h(n) of the prototype filter.c(n)=(−1)INT(n/2M)h(n) (0≦n≦N−1)  (formula 5)
Here, INT(x) is a function for making an integer with truncation of a fractional part of x.
In a conventional example of FIG. 13, by introducing an intermediate signal, it is possible to reduce quantity of arithmetic operation, as compared to a case of directly calculating (formula 2).
Here, quantity of arithmetic operation, which is necessary for realizing the complex exponential modulation signal analysis filter bank of FIG. 13, is evaluated with number of real number addition and number of real number multiplication.
In step 1301, assuming that c(n), which was calculated by (formula 5) in advance, is stored in a table and used, in lieu of h(n), the number of real number addition is (N/2M−1)(2M/K)=N/K−2M/K times, and the number of real number multiplication is (N/2M)(2M/K)=N/K times.
In step 1302, assuming that K exp(jπ(k+0.5)(2Kn+A)/(2M)) is calculated in advance and stored in a table to be used, the number of real number addition is 2(2M/K−1)(M/K)=4(M/K)2−2(M/K) times, and the number of real number multiplication is 2(2M/K)(M/K)=4(M/K)2 times.
Assuming that order N of the prototype filer is 640, and number of bands M is 64, and scaling factor K of up-sampling is 1, the number of real number addition is 512 times and the number of real number multiplication is 640 times, in step 1301, and the number of real number addition is 16256 times and the number of real number multiplication is 16384 times, in step 1302, and as a total of combination of step 1301 and step 1302, the number of real number addition is 16768 times and the number of real number multiplication is 17024 times.
Next, a conventional complex exponential modulation signal synthesis filter bank will be described. By the above-described patent publication document, filter coefficient hs(k, n) of n sample in k-th band of a complex exponential modulation signal synthesis filter bank is given by (formula 6) (B is phase for signal synthesis).hs(k,n)=(1/M)h(Ln)exp(jπ(k+0.5)(2Ln+B)/(2M))  (formula 6)
Here, phase B for signal synthesis satisfies a relational formula of (formula 7) (P is arbitrary integer) with phase A for signal analysis.A+B+2N=8MP  (formula 7)
A real number part of summation in an effective band (band from 0-th band up to (M/L−1)-th band) of such a signal that an input complex band signal is convolved with the filter coefficient of (formula 6) is an output of signal synthesis filter bank 1202. When values of filter coefficients of first and last prototype filters are set to zero, assuming that a complex band input signal at sample time mM/K in k-th band of a complex exponential modulation signal synthesis filter bank is X(k, mM/K), output signal x(mM/K+nL/K) at sampling time mM/K+nL/K is given by (formula 8) (Re(x) is an real part of complex number x).
                                                        x              ⁡                              (                                                      mM                    /                    K                                    +                                      nL                    /                    K                                                  )                                      =                                                  ⁢                                                  ⁢                          Re              ⁢                                                          ⁢                              {                                                      (                                          1                      /                      M                                        )                                    ⁢                                                                          ⁢                                                            ∑                                              k                        =                        0                                                                                              M                          /                          L                                                -                        1                                                              ⁢                                                                                  ⁢                                                                  ∑                                                  l                          =                          0                                                                                                      N                            /                            L                                                    -                          1                                                                    ⁢                                                                                          ⁢                                                                        X                          (                                                                                                          ⁢                                                      k                            ,                                                                                                                  ⁢                                                                                          mM                                /                                                                                                                                                                                                                    K                                      +                                                                                                                                                                                                                                          ⁢                                                                                                                          ⁢                                                              L                                (                                                                                                                                  ⁢                                                                  n                                  -                                                                                                                                          ⁢                                                                                                                                          ⁢                                                                                                                                                                                                                                                                           ⁢                                                                                               l                                )                                                                                                              )                                                ⁢                                                                                                  ⁢                                                  h                          (                                                                                                          ⁢                          Ll                          )                                                ⁢                                                                                                  ⁢                                                  exp                          (                                                                                                          ⁢                                                      j                            ⁢                                                                                                                  ⁢                                                          π                              ⁡                                                              (                                                                  k                                  +                                  0.5                                                                )                                                                                      ⁢                                                                                          (                                                                                                      2                                    ⁢                                    Ll                                                                    +                                  B                                                                )                                                            /                                                                                                                                 ⁢                                                  (                                                      2                            ⁢                            M                                                    )                                                                                                                    )                                              }                ⁢                                  ⁢                                  ⁢                  (                      0            ≦            n            ≦                                          M                /                L                            -              1                                )                                    (                  formula          ⁢                                          ⁢          8                )            
If (formula 8) is calculated directly, quantity of arithmetic operation becomes large, and therefore, in the same manner as in the case of the complex exponential modulation signal analysis filter bank, a complex exponential modulation signal synthesis bank with reduced quantity of arithmetic operation is used as follows, in prior art. FIG. 14 is a flow chart which shows processing steps of a conventional synthesizing method of a complex exponential modulation signal synthesis filter bank. In step 1401, intermediate signal w(n) of 0≦n≦2(N−M)/L−1 is shifted to w(n+2M/L), and intermediate signal w(n) of 0≦n≦2M/L−1 is calculated by (formula 9), from complex band input signal X(k, mM/K) at sampling time mM/K in k-th band.
                                          w            ⁡                          (              n              )                                =                      Re            ⁢                          {                                                (                                      1                    /                    M                                    )                                ⁢                                                      ∑                                          k                      =                      0                                                                                      M                        /                        L                                            -                      1                                                        ⁢                                                            X                      (                                                                                          ⁢                                              k                        ,                                                                                                  ⁢                                                  mM                          /                                                                                                          ⁢                          K                                                                    )                                        ⁢                                                                                  ⁢                                          exp                      (                                                                                          ⁢                                              j                        ⁢                                                                                                  ⁢                                                  π                          ⁡                                                      (                                                          k                              +                              0.5                                                        )                                                                          ⁢                                                                              (                                                                                          2                                ⁢                                L                                ⁢                                                                                                                                  ⁢                                n                                                            +                              B                                                        )                                                    /                                                      (                                                          2                              ⁢                              M                                                        )                                                                                              )                                                                                  }                                      ⁢                                  ⁢                                  ⁢                  (                      0            ≦            n            ≦                                          2                ⁢                                  M                  /                  L                                            -              1                                )                                    (                  formula          ⁢                                          ⁢          9                )            
Next, instep 1402, output signal x(mM/K+nL/K) at sampling time mM/K+nL/K (0≦n≦M/L−1) is calculated by (formula 10), from intermediate signal w(n).
                                          x            ⁡                          (                                                mM                  /                  K                                +                                  nL                  /                  K                                            )                                =                                    ∑                              i                =                0                                                              N                  /                                      (                                          2                      ⁢                      M                                        )                                                  -                1                                      ⁢                                                            (                                      -                    1                                    )                                i                            ⁢                              {                                                                            w                      ⁡                                              (                                                                              4                            ⁢                                                          Mi                              /                              L                                                                                +                          n                                                )                                                              ⁢                                          h                      ⁡                                              (                                                                              2                            ⁢                            Mi                                                    +                                                      L                            ⁢                                                                                                                  ⁢                            n                                                                          )                                                                              +                                                                          ⁢                                                            w                      ⁡                                              (                                                                              4                            ⁢                                                          Mi                              /                              L                                                                                +                                                      3                            ⁢                                                          M                              /                              L                                                                                +                          n                                                )                                                              ⁢                                          h                      ⁡                                              (                                                                              2                            ⁢                            Mi                                                    +                          M                          +                                                      L                            ⁢                                                                                                                  ⁢                            n                                                                          )                                                                                            }                                                    ⁢                                  ⁢                                  ⁢                  (                      0            ≦            n            ≦                                          M                /                L                            -              1                                )                                    (                  formula          ⁢                                          ⁢          10                )            
Pages 60, 61, and 63 of the above-described ISO/IEC document describe an example of a complex exponential modulation signal synthesis filter bank for such a case that number of bands M is 64, and filter order N of a prototype filter is 640, and scaling factor K of up-sampling is 2, and scaling factor 1/L of down-sampling is 1, and phase B for signal synthesis is −255.
Quantity of arithmetic operation, which is necessary for realizing the complex exponential modulation signal synthesis filter bank of FIG. 14, is evaluated with number of real number addition and number of real number multiplication.
In step 1401, assuming that (1/M)exp(jπ(k+0.5)(2Ln+B)/(2M)) is calculated in advance, and stored in a table to be used, the number of real number addition is (M/L−1)(2M/L)+(M/L)(2M/L)=4(M/L)2−2(M/L) times, and the number of real number multiplication is 2(M/L)(2M/L)=4(M/L)2 times.
In step 1402, assuming that c(n), which was calculated by (formula 5), is stored in a table and used, in lieu of h(n), the number of real number addition is (N/M−1)(M/L)=N/L−M/L times, and the number of real number multiplication is (N/2M)2(M/L)=N/L times.
Assuming that order N of the prototype filer is 640, and number of bands M is 64, and scaling factor 1/L of down-sampling is 1, the number of real number addition is 16256 times and the number of real number multiplication is 16384 times, in step 1401, and the number of real number addition is 576 times and the number of real number multiplication is 640 times, in step 1402, and as a total of combination of step 1401 and step 1402, the number of real number addition is 16832 times and the number of real number multiplication is 17024 times.